The purpose of the proposed psychological research is to study the ability of trainable (TMR) and educable mentally retarded (EMR) children to learn basic mathematical skills and concepts, which have usually been assumed to be unlearnable by these populations. The project will attempt to encourage and measure meaningful learning rather than conditioned responses or rote associations. The congnitive approach and analysis will focus on genuine understanding, rule-governed skills, and problem solving. Such higher-order learning of basic skills and concepts is a prerequisite for further math instruction and transfer to survival skills (e.g., time-telling, making change, measuring quantities, etc.). The project consists of a series of four TMR and five EMR training studies extending over five years' time. One focus is counting skills (e.g., rule-governed production of the number sequence, spontaneous and appropriate application of object counting, and the abstraction of regulatiries about number from counting experience) and number sense (e.g., using number mentally to compare sets). Basic numeration skills and abilities (e.g., reading and writing numerals, an appreication of place value and zero as a place holder) will also be examined. The third focus is the development of basic arithmetic abilities: performing addition and subtraction mentally and with written symbols. Of particular interest is whether or not TMR and EMR childrent can invent more economical computing procedures, abstract such regulatiries as the commutativity of addition, internalize and generalize such regularities as N + O = N, and understand and apply the complementary relationship between addition and subtraction. Half of the approximately 30 subjects in each study will be randomly assigned to an experimental group; half to a control group. Subjects will be matched on pretest performance. An immediate posttest and a delayed posttest will be used. Learning will be measured rigorously in terms of transfer to new problems and everyday situations as well as in terms of long-term retention.